Selections and selectors

The aim of the paper is to outline the known results and the main technics they are obtained with as well as the open questions about selections of set-valued mappings. More precisely the following topics will be touched: Set-valued selections of set-valued mappings, factorizations of set-valued mappings, inverse selection theorems, selections via extension of the set-valued mapping.     

On hyperspace topologies via distance functionals in quasi-metric spaces

Summary Let (X,d) be a quasi-pseudo-metric space. We investigate hyperspace topologies on P₀(X) defined by distance functionals. In particular, the K-topology is introduced and compared with other hyperspace topologies. Some properties of the Wijsman topology and the K-topology are explored.     

Selections, extensions and collectionwise normality

We demonstrate that the classical Michael selection theorem for l.s.c. mappings with a collectionwise normal domain can be reduced only to compact-valued mappings modulo Dowker's extension theorem for such spaces. The idea used to achieve this reduction is also applied to get a simple direct proof of that selection theorem of Michael's. Some other possible applications are demonstrated as well.     

Baire spaces and hyperspace topologies

Sufficient conditions for abstract (proximal) hit-and-miss hyperspace topologies and the Wijsman hyperspace topology, respectively, are given to be Baire spaces, thus extending results of McCoy, Beer, and Costantini. Further the quasi-regularity of (proximal) hit-and-miss topologies is investigated.

     

Selections and countably-approachable points

The present paper improves a result of Gutev [V. Gutev, Approaching points by continuous selections, J. Math. Soc. Japan (4) (2006) 1203–1210] by characterizing the countably-approachable points in sense of [V. Gutev, Approaching points by continuous selections, J. Math. Soc. Japan (4) (2006) 1203–1210] by a natural extreme-like condition in the spirit of [V. Gutev, T. Nogura, Vietoris continuous selections and disconnectedness-like properties, Proc. Amer. Math. Soc. 129 (2001) 2809–2815; V. Gutev, T. Nogura, Selection pointwise-maximal spaces, Topology Appl. 146–147 (2005) 397–408]. This demonstrates the natural relationship between different extreme-like points with respect to continuous selections for the Vietoris hyperspace of nonempty closed subsets.     

Weak topology and Browder-Kirk's theorem on hyperspace

Let K be a weakly compact, convex subset of a Banach space X with normal structure. Browder-Kirk's theorem states that every non-expansive mapping T which maps K into K has a fixed point in K. Suppose now that WCC(X) is the collection of all non-empty weakly compact convex subsets of X. We shall define a certain weak topology Tw on WCC(X) and have the above-mentioned result extended to the hyperspace (WCC(X);Tw).     

Fell topology on the hyperspace of a non-Hausdorff space

Fell topology is very widely used today, even in metric spaces; but J. Fell introduced it in a non-Hausdorff context in the connection with the theory of C ^*-algebras. In spite of this, it has been studied only on the hyperspace of a Hausdorff space, except for the first results due to Fell himself. The present paper aims to fill this gap, in particular extending some results of H. Poppe and of G. Beer to the general case. Project 10251002 supported by National Natural Science Foundation of China.     

Group topologies coarser than the Isbell topology

The Isbell, compact-open and point-open topologies on the set C(X,R) of continuous real-valued maps can be represented as the dual topologies with respect to some collections α(X) of compact families of open subsets of a topological space X. Those α(X) for which addition is jointly continuous at the zero function in Cα(X,R) are characterized, and sufficient conditions for translations to be continuous are found. As a result, collections α(X) for which Cα(X,R) is a topological vector space are defined canonically. The Isbell topology coincides with this vector space topology if and only if X is infraconsonant. Examples based on measure theoretic methods, that Cα(X,R) can be strictly finer than the compact-open topology, are given. To our knowledge, this is the first example of a splitting group topology strictly finer than the compact-open topology.     

Selection principles and hyperspace topologies

In this paper we investigate relationships between closure-type properties of hyperspaces over a space X and covering properties of X.     

集值映射的连续选择与线性算子的齐性右逆

本文对于取值在Ｂａｎａｃｈ空间的几乎下半连续映射引入两个相关的下半连续闭凸集值映射,得到集值映射的连续选择存在性的若干特征,从而将 Ｄｅｕｔｓｃｈ Ｅ,和Ｋｅｎｄｅｒｏｖ Ｐ,Ｄｅ Ｂｌａｓｔ Ｆ．Ｓ．和 Ｍｙｊａｋ Ｊ,Ｐｒｚｅｓｌａｗｓｋｉ Ｋ．和 Ｒｙｂｉｎｓｋｉ Ｌ Ｅ,Ｇｕｔｅｖ Ｖ．等人以及作者自己的关于连续选择存在性的结果作为推论给出．并用这些结论讨论了在开映射定理不成立的。情况下从Ｂａｎａｃｈ空间到赋范空间上线性连续算子的齐性右逆存在问题．     

Distance functional and suprema of hyperspace topologies

Let CL(X) denote the nonempty closed subsets of a metrizable space X. We show that the Vietoris topology on CL(X) is the weakest topology on CL(X) such that A - d(x, A) is continuous for each x X and each admissible metric d. We also give a concrete presentation of the analogous weak topology for uniformly equivalent metrics, and are led to consider for an admissible metric d the weakest topology on CL(X) such that the gap functional (A, B) - {d(ta, b): a A, b B} is continuous on CL(X) × CL(X).Visiting the University of Minnesota.Visiting California State University, Los Angeles.     

Local dendrites with unique n-fold hyperspace

Let Z be a metric continuum and n be a positive integer. Let Cn(Z) be the hyperspace of the nonempty closed subsets of Z with at most n components. In this paper we prove the following result: Let X be a local dendrite such that every point of X has a neighborhood which is a dendrite whose set of end points is closed and Z is any continuum such that Cn(X) is homeomorphic to Cm(Z) for some n,m?3, then X is homeomorphic to Z.     

超空间2^X的某些性质和应用

本文分别改进了文献[1]和[2]的一个结果,并且应用超空间的某些性质,把C.Conley定理推广到拓扑群.     

Set-maximal selections

The present paper deals with continuous selections f for the Vietoris hyperspace of all nonempty closed subsets of a space X which are maximal with respect to some closed subset SX, i.e. with the property that f(T)S for every with T∩S≠. This gives rise to a common point of view of several extreme-like properties studied before, it also provides an useful tool in classifying disconnectedness-like properties of spaces by continuous selections for the Vietoris hyperspace.     

赋予hit—or-miss拓扑超空间动力系统的拓扑熵

设（X,d,f）为拓扑动力系统,其中X为局部紧第二可数Hausdorff空间,d为紧型度量,f为完备映射,用2^x和f分别表示由X的所有非空闭子集和所有闭子集构成的集族,（2^x,ρ,2^f）和（f,ρ,2^f）为由（X,d,f）诱导的赋予hit—or—miss拓扑的超空间动力系统．本文研究了h（X,d,f）和h（2^...     

Functorial topologies and finite-index subgroups of abelian groups

In the general context of functorial topologies, we prove that in the lattice of all group topologies on an abelian group, the infimum between the Bohr topology and the natural topology is the profinite topology. The profinite topology and its connection to other functorial topologies is the main objective of the paper. We are particularly interested in the poset C(G) of all finite-index subgroups of an abelian group G, since it is a local base for the profinite topology of G. We describe various features of the poset C(G) (its cardinality, its cofinality, etc.) and we characterize the abelian groups G for which C(G)?{G} is cofinal in the poset of all subgroups of G ordered by inclusion. Finally, for pairs of functorial topologies T, S we define the equalizer E(T,S), which permits to describe relevant classes of abelian groups in terms of functorial topologies.     

Coincidence of the Isbell and fine Isbell topologies

Let C(X,Y) be the set of all continuous functions from a topological space X into a topological space Y. We find conditions on X that make the Isbell and fine Isbell topologies on C(X,Y) equal for all Y. For zero-dimensional spaces X, we show there is a space Z such that the coincidence of the Isbell and fine Isbell topologies on C(X,Z) implies the coincidence on C(X,Y) for all Y. We then consider the question of when the Isbell and fine Isbell topologies coincide on the set of continuous real-valued functions. Our results are similar to results established for consonant spaces.     

On the Existence of Continuous Selections in Subset Spaces with Ochan-Type Topologies

We prove the existence of a continuous selection in the space (X) (the nonempty subsets of a topologic space X) equipped with the Ochan-type -topology in the case where X can be imbedded into the Tikhonov product of a countable family of left spaces (Theorem 1). We also prove the existence of a continuous selection on the exponent of a complete metric space in the Ochan-type -topology (Theorem 2).     

Sections, selections and Prohorov's theorem

The famous Prohorov theorem for Radon probability measures is generalized in terms of usco mappings. In the case of completely metrizable spaces this is achieved by applying a classical Michael result on the existence of usco selections for l.s.c. mappings. A similar approach works when sieve-complete spaces are considered.